Elaborate methodologies have been developed to study the thermo-chemical response of materials in high-enthalpy flows. To reach the high magnitudes of heat flux encountered in some hypersonic applications, one can resort to supersonic jets. They involve several physical effects, such as detached shocks ahead of probes. Because of these features, characterizing supersonic flows is a challenging task, especially when one accounts for experimental and modeling uncertainties. Building on the development of stochastic approaches, we propose a holistic methodology to determine the quantities of interest in an optimal manner for an under-expanded high-enthalpy jet, using both experimental measurements and high-fidelity flow simulations. Given the high computational cost of the high-fidelity simulations needed to describe the flow, we built an adaptive/multi-fidelity surrogate model to replace the estimation of the costly computer solver. A Bayesian inference method then allowed for characterizing an experiment carried out in the von Karman Institute's Plasmatron facility, for which no robust methodology currently exists. We show that the reservoir pressure and temperature and the nitrogen catalytic recombination coefficient of the copper probes can be accurately determined from the available measurements. Contrarily, the test conditions do not allow us to estimate the oxygen catalytic recombination coefficient. Finally, the characterized uncertainties are propagated through the numerical solver, yielding an uncertainty-based high-fidelity representation of the hypersonic flow's structure variability.

1.
V. I.
Sakharov
, “
Numerical simulation of thermally and chemically nonequilibrium flows and heat transfer in underexpanded induction plasmatron jets
,”
Fluid Dyn.
42
,
1007
1168
(
2007
).
2.
A.
Gordeev
,
A.
Kolesnikov
, and
V.
Sakharov
, “
Flow and heat transfer in underexpanded nonequilibrium jets of an induction plasmatron
,”
Fluid Dyn.
46
,
623
633
(
2011
).
3.
D. Le
Quang
,
Y.
Babou
,
O.
Chazot
, and
P.
Andre
, “
Investigation of supersonic air plasma jet produced in the VKI Plasmatron facility
,”
ESA Spec. Publ.
714
,
31
(
2012
).
4.
C.
Purpura
,
F.
De Filippis
,
P.
Barrera
, and
D.
Mandanici
, “
Experimental characterisation of the cira plasma wind tunnel Scirocco test section
,”
Acta Astronaut.
62
,
410
421
(
2008
).
5.
U.
Duzel
,
O.
Schroeder
,
H.
Zhang
, and
A.
Martin
, “
Numerical simulation of an arc jet test section
,”
J. Thermophys. Heat Transfer
34
,
393
403
(
2020
).
6.
A.
Balter-Peterson
,
F.
Nichols
,
B.
Mifsud
, and
W.
Love
, “
Arc jet testing in NASA AMES Research Center thermophysics facilities
,” in
AlAA 4th International Aerospace Planes Conference
(
American Institute of Aeronautics and Astronautics
,
1992
).
7.
E.
Franquet
,
V.
Perrier
,
S.
Gibout
, and
P.
Bruel
, “
Free underexpanded jets in a quiescent medium: A review
,”
Prog. Aerosp. Sci.
77
,
25
53
(
2015
).
8.
R.
Muraoka
and
T.
Hiejima
, “
Onset conditions for Mach disk formation in underexpanded jet flows
,”
Phys. Fluids
34
,
116125
(
2022
).
9.
C.
Park
,
G.
Raiche
,
D.
Driver
,
J.
Olejniczak
,
I.
Terrazas-Salinas
,
T.
Hightower
, and
T.
Sakai
, “
Comparison of enthalpy determination methods for an arc-jet facility
,”
J. Thermophys. Heat Transfer
20
,
672
679
(
2006
).
10.
C.
Shepard
,
F.
Milos
, and
J.
Taunk
, “
A sonic flow equation for electric arc jets
,” in
24th Plasma Dynamics, and Lasers Conference
(
American Institute of Aeronautics and Astronautics
,
1993
).
11.
R.
Pope
, “
Measurements of enthalpy in low-density arc-heated flows
,”
AIAA J.
6
,
103
110
(
1968
).
12.
J.
Fay
and
F.
Riddell
, “
Theory of stagnation point heat transfer in dissociated air
,”
J. Aerosp. Sci.
25
,
73
85
(
1958
).
13.
R.
Goulard
, “
On catalytic recombination rates in hypersonic stagnation heat transfer
,”
AIAA J.
28
,
737
745
(
1958
).
14.
A.
Fagnani
,
D.
Le Quang Huy
,
B.
Helber
,
S.
Demange
,
A.
Turchi
,
O.
Chazot
, and
A.
Hubin
, “
Investigation of a free-stream air plasma flow by optical emission spectroscopy and comparison to magnetohydrodynamics simulations
,” in
AIAA SciTech 2020 Forum
(
American Institute of Aeronautics and Astronautics
,
2020
).
15.
D.
LeQuang
, “
Spectroscopic measurements of sub- and supersonic plasma flows for the investigation of atmospheric re-entry shock layer radiation
,” Ph.D. thesis (
Université Blaise Pascal
,
2014
).
16.
D.
Prabhu
,
D.
Saunders
,
T.
Oishi
,
K.
Skokova
,
J.
Santos
,
J.
Fu
,
I.
Terrazas-Salinas
,
J.
Carballo
, and
D.
Driver
, “
CFD analysis framework for arc-heated flowfields. I: Stagnation testing in arc-jets at NASA ARC
,” in
41st AIAA Thermophysics Conference
(
American Institute of Aeronautics and Astronautics
,
2009
).
17.
O.
Chazot
, “
Experimental studies on hypersonic stagnation point chemical environment
,”
Tech. Rep.
RTO-EN-AVT-142 (
2006
).
18.
T.
Gökçen
,
G.
Raiche
,
D.
Driver
,
J.
Balboni
, and
R.
McDaniel
, “
Applications of CFD analysis in arc-jet testing of RCC plug repairs
,” in
25th AIAA Aerodynamic Measurement Technology and Ground Testing Conference
(
American Institute of Aeronautics and Astronautics
,
2006
).
19.
A.
Nawaz
,
D.
Driver
,
I.
Terrazas-Salinas
, and
S.
Sepka
, “
Surface catalysis and oxidation on stagnation point heat flux measurements in high enthalpy arc jets
,” in
44th AIAA Thermophysics Conference
(
American Institute of Aeronautics and Astronautics
,
2013
).
20.
A.
Viladegut
and
O.
Chazot
, “
Empirical modeling of copper catalysis for enthalpy determination in plasma facilities
,”
J. Thermophys. Heat Transfer
34
,
26
36
(
2020
).
21.
A.
Turchi
,
P.
Congedo
, and
T.
Magin
, “
Thermochemical ablation modeling forward uncertainty analysis—Part I: Numerical methods and effect of model parameters
,”
Int. J. Therm. Sci.
118
,
497
509
(
2017
).
22.
A.
Turchi
,
P.
Congedo
,
B.
Helber
, and
T.
Magin
, “
Thermochemical ablation modeling forward uncertainty analysis—Part II: Application to plasma wind-tunnel testing
,”
Int. J. Therm. Sci.
118
,
510
517
(
2017
).
23.
F.
Sanson
,
F.
Panerai
,
T.
Magin
, and
P.
Congedo
, “
Robust reconstruction of the catalytic properties of thermal protection materials from sparse high-enthalpy facility experimental data
,”
Exp. Therm. Fluid Sci.
96
,
482
(
2018
).
24.
A.
del Val
,
O.
Le Maître
,
P.
Congedo
, and
T.
Magin
, “
Stochastic calibration of a carbon nitridation model from plasma wind tunnel experiments using a Bayesian formulation
,”
Carbon
200
,
199
214
(
2022
).
25.
A.
del Val
,
T.
Magin
, and
P.
Congedo
, “
Quantification of model-form uncertainties affecting the calibration of a carbon nitridation model by means of Bayesian model averaging
,”
Int. J. Heat Mass Transfer
213
,
124271
(
2023
).
26.
A.
del Val
,
O.
Le Maître
,
T.
Magin
,
O.
Chazot
, and
P.
Congedo
, “
A surrogate-based optimal likelihood function for the Bayesian calibration of catalytic recombination in atmospheric entry protection materials
,”
Appl. Math. Modell.
101
,
791
810
(
2022
).
27.
P.
Ventura Diaz
,
A.
Parente
,
J.
Meurisse
,
S.
Yoon
, and
N.
Mansour
, “
High-fidelity numerical analysis of arc-jet aerothermal environments
,” in
AIAA SciTech 2020 Forum
(
American Institute of Aeronautics and Astronautics
,
2020
).
28.
A.
Brune
,
T.
West
, and
L.
White
, “
Calibration probe uncertainty and validation for the hypersonic material environmental test system
,”
J. Thermophys. Heat Transfer
34
,
404
420
(
2020
).
29.
D.
Xiu
and
G.
Karniadakis
, “
The Wiener–Askey polynomial chaos for stochastic differential equations
,”
SIAM J. Sci. Comput.
24
,
619
644
(
2002
).
30.
J.
Sacks
,
W.
Welch
,
T.
Mitchell
, and
H.
Wynn
, “
Design and analysis of computer experiments
,”
Stat. Sci.
4
,
409
423
(
1989
).
31.
A.
Forrester
,
A.
Sobester
, and
A.
Keane
,
Engineering Design via Surrogate Modelling: A Practical Guide
(
Wiley
,
2008
).
32.
B.
Peherstorfer
,
K.
Willcox
, and
M.
Gunzburger
, “
Survey of multifidelity methods in uncertainty propagation, inference, and optimization
,”
SIAM Rev.
60
,
550
591
(
2018
).
33.
M.
Giselle Fernández-Godino
,
C.
Park
,
N.
Kim
, and
R.
Haftka
, “
Issues in deciding whether to use multifidelity surrogates
,”
AIAA J.
57
,
2039
2054
(
2019
).
34.
Z.
Han
,
C.
Xu
,
L.
Zhang
,
Y.
Zhang
,
K.
Zhang
, and
W.
Song
, “
Efficient aerodynamic shape optimization using variable-fidelity surrogate models and multilevel computational grids
,”
Chin. J. Aeronaut.
33
,
31
47
(
2020
).
35.
I.
Abdallah
,
C.
Lataniotis
, and
B.
Sudret
, “
Parametric hierarchical kriging for multi-fidelity aero-servo-elastic simulators—Application to extreme loads on wind turbines
,”
Probabilistic Eng. Mech.
55
,
67
77
(
2019
).
36.
H.
Babaee
,
P.
Perdikaris
,
C.
Chryssostomidis
, and
G. E.
Karniadakis
, “
Multi-fidelity modelling of mixed convection based on experimental correlations and numerical simulations
,”
J. Fluid Mech.
809
,
895
917
(
2016
).
37.
L.
Zheng
,
T.
Hedrick
, and
R.
Mittal
, “
A multi-fidelity modeling approach for evaluation and optimization of wing stroke aerodynamics in flapping flight
,”
J. Fluid Mech.
721
,
118
154
(
2013
).
38.
A.
Sadagopan
,
D.
Huang
,
U.
Duzel
,
L.
Martin
, and
K.
Hanquist
, “
Assessment of high-temperature effects on hypersonic aerothermoelastic analysis using multi-fidelity multi-variate surrogates
,” in
AIAA SciTech 2021 Forum
(
American Institute of Aeronautics and Astronautics
,
2021
).
39.
M.
Santos
,
S.
Hosder
, and
T.
West
, “
Multifidelity turbulent heating prediction of hypersonic inflatable aerodynamic decelerators with surface scalloping
,”
J. Spacecr. Rockets
58
,
1325
1338
(
2021
).
40.
K.
Quinlan
,
J.
Movva
,
E.
Stein
, and
A.
Kupresanin
, “
Leveraging multi-fidelity aerodynamic databasing to efficiently represent a hypersonic design space
,” in
Ascend 2021
(
American Institute of Aeronautics and Astronautics
,
2021
).
41.
M.
Kennedy
and
A.
O'Hagan
, “
Predicting the output from a complex computer code when fast approximations are available
,”
Biometrika
87
,
1
13
(
2000
).
42.
A.
Forrester
,
A.
Sobester
, and
A.
Keane
, “
Multi-fidelity optimization via surrogate modelling
,”
Proc. R. Soc. A
463
,
3251
3269
(
2007
).
43.
Z.
Han
and
S.
Görtz
, “
Hierarchical kriging model for variable-fidelity surrogate modeling
,”
AIAA J.
50
,
1885
1896
(
2012
).
44.
V.
Picheny
,
T.
Wagner
, and
D.
Ginsbourger
, “
A benchmark of kriging-based infill criteria for noisy optimization
,”
Struct. Multidisc. Optim.
48
,
607
(
2013
).
45.
B.
Bottin
,
O.
Chazot
,
M.
Carbonaro
,
V.
Haegen
, and
S.
Paris
, “
The VKI Plasmatron characteristics and performance
,”
Tech. Rep. ADPO10745
(
2000
).
46.
A.
Fagnani
,
B.
Helber
,
A.
Hubin
, and
O.
Chazot
, “
Line-of-sight gas radiation effects on near-infrared two-color ratio pyrometry measurements during plasma wind tunnel experiments
,”
Measurement
227
,
114175
(
2024
).
47.
B.
Helber
, “
Material response characterization of low-density ablators in atmospheric entry plasmas
,” Ph.D. thesis (
Vrije Universiteit Brussel
,
2016
).
48.
F.
Panerai
, “
Aerothermochemistry characterization of thermal protection systems
,” Ph.D. thesis (
Université Degli Studi Di Perugia
,
2012
).
49.
A.
Viladegut
and
O.
Chazot
, “
Catalytic characterization in plasma wind tunnels under the influence of gaseous recombination
,”
Phys. Fluids
34
,
027108
(
2022
).
50.
A.
Fagnani
,
B.
Dias
,
P.
Schrooyen
,
B.
Helber
,
T.
Magin
, and
O.
Chazot
, “
Investigation of quartz ablation in the VKI Plasmatron Facility: Comparison between experimental and numerical results
,” in
AIAA Aviation 2021 Forum
(
American Institute of Aeronautics and Astronautics
,
2021
).
51.
B.
Helber
,
A.
Turchi
, and
T.
Magin
, “
Determination of active nitridation reaction efficiency of graphite in inductively coupled plasma flows
,”
Carbon
125
,
582
594
(
2017
).
52.
B.
Helber
,
O.
Chazot
,
A.
Hubin
, and
T.
Magin
, “
Microstructure and gas-surface interaction studies of a low-density carbon-bonded carbon fiber composite in atmospheric entry plasmas
,”
Compos. Part A: Appl. Sci. Manuf.
72
,
96
107
(
2015
).
53.
A.
Cipullo
,
B.
Helber
,
F.
Panerai
,
L.
Zeni
, and
O.
Chazot
, “
Investigation of freestream plasma flow produced by inductively coupled plasma wind tunnel
,”
J. Thermophys. Heat Transfer
28
,
381
393
(
2014
).
54.
F.
Panerai
and
O.
Chazot
, “
Characterization of gas/surface interactions for ceramic matrix composites in high enthalpy, low pressure air flow
,”
Mater. Chem. Phys.
134
,
597
607
(
2012
).
55.
D.
Guariglia
,
B.
Helber
, and
O.
Chazot
, “
Very high heat-flux measurements in plasmatron with subsonic and supersonic plasma flow
,” in
8th European Symposium on Aerothermodynamics for Space Vehicles
(
European Space Agency
,
2015
).
56.
B.
Helber
,
A.
Turchi
,
O.
Chazot
,
T.
Magin
, and
A.
Hubin
, “
Gas/surface interaction study of low-density ablators in sub- and supersonic plasmas
,” in
11th AIAA/ASME Joint Thermophysics and Heat Transfer Conference
(
American Institute of Aeronautics and Astronautics
,
2014
).
57.
N.
Metropolis
,
A.
Rosenbluth
,
M.
Rosenbluth
,
A.
Teller
, and
E.
Teller
, “
equation of state calculations by fast computing machines
,”
J. Chem. Phys.
21
,
1087
1092
(
1953
).
58.
W.
Hastings
, “
Monte Carlo sampling methods using Markov chains and their applications
,”
Biometrika
57
,
97
109
(
1970
).
59.
A.
Padron
,
J.
Alonso
,
F.
Palacios
,
M.
Barone
, and
M.
Eldred
, “
Multi-fidelity uncertainty quantification: Application to a vertical axis wind turbine under an extreme gust
,” in
15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
(
American Institute of Aeronautics and Astronautics
,
2014
).
60.
L.
Ng
and
M.
Eldred
, “
Multifidelity uncertainty quantification using non-intrusive polynomial chaos and stochastic collocation
,” in
53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference
(
American Institute of Aeronautics and Astronautics
,
2012
).
61.
M.
Eldred
, “
Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design
,” in
50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
(
American Institute of Aeronautics and Astronautics
,
2009
).
62.
Y.
Zhang
,
Z.
Han
, and
k
Zhang
, “
Variable-fidelity expected improvement method for efficient global optimization of expensive functions
,”
Struct. Multidisc. Optim.
58
,
1431
(
2018
).
63.
S.
Marelli
and
B.
Sudret
, “
UQLab: A framework for uncertainty quantification in Matlab
,” in
Vulnerability, Uncertainty, and Risk
(
American Society of Civil Engineers
,
2014
), pp.
2554
2563
.
64.
C.
Lataniotis
,
S.
Marelli
, and
B.
Sudret
, “
The Gaussian Process Modeling Module in UQLab
,” (
2018
).
65.
S.
Gordon
and
J.
McBride
, “
Thermodynamic data to 20,000 K for monatomic gases
,”
Tech. Rep. 1999-208523
(
1999
).
66.
J.
Ramshaw
, “
Self-consistent effective binary interaction approximation for strongly coupled multifluid dynamics
,”
J. Non-Equilibrium Thermodyn.
23
,
135
140
(
1998
).
67.
R.
Gupta
,
J.
Yos
,
R.
Thompson
, and
K.
Lee
, “
A review of reaction rates and thermodynamic and transport properties for an 11-species air model for chemical and thermal nonequilibrium calculations to 30000 K
,”
Tech. Rep. NASA-RP-1232
(
1990
).
68.
C.
Park
,
R.
Jaffe
, and
H.
Partridge
, “
Chemical-kinetic parameters of hyperbolic earth entry
,”
J. Thermophys. Heat Transfer
15
,
76
90
(
2001
).
69.
G.
Bellas-Chatzigeorgis
,
A.
Turchi
,
A.
Viladegut
,
O.
Chazot
,
P.
Barbante
, and
T.
Magin
, “
Development of catalytic and ablative gas-surface interaction models for the simulation of reacting gas mixtures
,” in
23rd AIAA Computational Fluid Dynamics Conference
(
American Institute of Aeronautics and Astronautics
,
2017
).
70.
J.
Scoggins
,
V.
Leroy
,
G.
Bellas-Chatzigeorgis
,
B.
Dias
, and
T.
Magin
, “
Mutation++: Multicomponent thermodynamic and transport properties for ionized gases in C++
,”
SoftwareX
12
,
100575
(
2020
).
71.
G.
Candler
,
H.
Johnson
,
I.
Nompelis
,
V.
Gidzak
,
P.
Subbareddy
, and
M.
Barnhardt
, “
Development of the US3D code for advanced compressible and reacting flow simulations
,” in
53rd AIAA Aerospace Sciences Meeting
(
American Institute of Aeronautics and Astronautics
,
2015
).
72.
J.
Steger
and
R.
Warming
, “
Flux vector splitting of the inviscid gas dynamic equations with application to finite-difference methods
,”
J. Comput. Phys.
40
,
263
293
(
1981
).
73.
M.
Wright
,
G.
Candler
, and
M.
Prampolini
, “
Data-parallel lower-upper relaxation method for the Navier-Stokes equations
,”
AIAA J.
34
,
1371
1377
(
1996
).
74.
M.
Capriati
,
K.
Prata
,
T.
Schwartzentruber
,
G.
Candler
, and
T.
Magin
, “
Development of a nitridation gas-surface boundary condition for high-fidelity hypersonic simulations
,” in
14th WCCM-ECCOMAS Congress
(
CIMNE
,
2021
).
75.
A.
Baskaya
,
M.
Capriati
,
D.
Ninni
,
F.
Bonelli
,
G.
Pascazio
,
A.
Turchi
,
T.
Magin
, and
S.
Hickel
, “
Verification and validation of immersed boundary solvers for hypersonic flows with gas-surface interactions
,” in
AIAA Aviation 2022 Forum
(
American Institute of Aeronautics and Astronautics
,
2022
).
76.
G.
Bellas-Chatzigeorgis
, “
Development of advanced gas-surface interaction models for chemically reacting flows for re-entry conditions
,” Ph.D. thesis (
Politecnico di Milano
,
2018
).
77.
L.
Eça
and
M.
Hoekstra
, “
A procedure for the estimation of the numerical uncertainty of CFD calculations based on grid refinement studies
,”
J. Comput. Phys.
262
,
104
130
(
2014
).
78.
S.
Kumar
and
A.
Assam
, “
Effect of rarefaction on thermal and chemical non-equilibrium for hypersonic flow with different enthalpy and catalytic wall conditions
,”
J. Thermal Sci. Eng. Appl.
15
,
071012
(
2023
).
79.
Y.
Prevereaud
,
J.
Vérant
, and
J.
Annaloro
, “
Noncatalytic and finite catalytic heating models for atmospheric re-entry codes
,” in
First International Orbital Debris Conference (IOC)
(
Lunar and Planetary Institute
,
Sugar Land, TX
,
2019
).
80.
K.
Sutton
and
R.
Graves
, “
A general stagnation-point convective heating equation for arbitrary gas mixtures
,”
Tech. Rep. 19720003329
(
1971
).
81.
A.
Munafò
and
T.
Magin
, “
Modeling of stagnation-line nonequilibrium flows by means of quantum based collisional models
,”
Phys. Fluids
26
,
097102
(
2014
).
82.
A.
Gelman
and
D.
Rubin
, “
Inference from iterative simulation using multiple sequences
,”
Stat. Sci.
7
,
457
472
(
1992
).
83.
S.
Brooks
and
A.
Gelman
, “
General methods for monitoring convergence of iterative simulations
,”
J. Comput. Graphical Stat.
7
,
434
455
(
1998
).
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